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Some very basic questions on topology!

Mathelogician

Member
Aug 6, 2012
35
hello everybody!
My knowledge in the field is so basic. So please answer as clear and simple as possible:
Answering to any one (even not all!) would be helpful.
The following 4 questions are the ones i couldn't solve after trying!
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1-Prove [0,1] is not isometric to [0,2].
2-For which intervals [a,b] in R is the intersection of [a,b] and Q a clopen (closed and open) subset of the metric space Q?
3-Find a metric space in which the boundary of Mr(p)={xEM :d(x,p)< r} is not equal to the sphere of radius r at p ,{xEM : d(x,p)=r}.
4- For a subspace N of a metric space M,prove that if the openness of S c N is equivalent to openness of S in M, then N is open in M.
============================================================

The next is about my proof of the following problem:

Let (An) be a nested decreasing sequence of non-empty closed sets in the metric space M.
If M is complete(every Cauchy is convergent) and diam(An) -> 0 as n -> infinity, show that S=Intersection of (An) is exactly one point( note that diam X= sup{d(x,y): x,y E X} ).
My proof : (I use Euclidean metric on R for simplicity of the proof- it can be done for any other metric on R)
Lets x,y E S. we must show that d(x,y)=0.
Lets d(x,y)= t > 0. By the red part: there exists a N1 such that for all n>N1 , we have diam(An)<t
Let k > n. So diam(Ak)<t.
In other hand, d(x,y) =< diam(Ak)< t. So d(x,y)< t which contradicts the blue.

Now my question is : What was the usage of supposing Ai's as Closed subspaces of M and M's being complete, in this proof?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,707
Re: Some very basic quastions on topology!

hello everybody!
My knowledge in the field is so basic. So please answer as clear and simple as possible:
Answering to any one (even not all!) would be helpful.
The following 4 questions are the ones i couldn't solve after trying!
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1-Prove [0,1] is not isometric to [0,2]. The interval [0,2] contains two points whose distance apart is 2, but the interval [0,1] does not.
2-For which intervals [a,b] in R is the intersection of [a,b] and Q a clopen (closed and open) subset of the metric space Q? Think about what difference it makes whether a and b are rational or irrational.
3-Find a metric space in which the boundary of Mr(p)={xEM :d(x,p)< r} is not equal to the sphere of radius r at p ,{xEM : d(x,p)=r}. Think about a space with a discrete metric.
4- For a subspace N of a metric space M, prove that if the openness of S c N is equivalent to openness of S in M, then N is open in M. What happens if you take S = N?
============================================================
(See the hints in red.)


The next is about my proof of the following problem:
Let (An) be a nested decreasing sequence of non-empty closed sets in the metric space M.
If M is complete(every Cauchy is convergent) and diam(An) -> 0 as n -> infinity, show that S=Intersection of (An) is exactly one point( note that diam X= sup{d(x,y): x,y E X} ).
My proof : (I use Euclidean metric on R for simplicity of the proof- it can be done for any other metric on R)
Lets x,y E S. we must show that d(x,y)=0.
Lets d(x,y)= t > 0. By the red part: there exists a N1 such that for all n>N1 , we have diam(An)<t
Let k > n. So diam(Ak)<t.
In other hand, d(x,y) =< diam(Ak)< t. So d(x,y)< t which contradicts the blue.

Now my question is : What was the usage of supposing Ai's as Closed subspaces of M and M's being complete, in this proof?
Your proof shows that S cannot contain more than one point. But how do you know that it contains any points at all? Answer: You need those extra conditions to ensure that S is not empty.